angel durán, denys dutykh, dimitrios mitsotakis

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Peregrine’s system revisitedAngel Durán, Denys Dutykh, Dimitrios Mitsotakis

To cite this version:Angel Durán, Denys Dutykh, Dimitrios Mitsotakis. Peregrine’s system revisited. Abcha N., PelinovskyE., Mutabazi I. Nonlinear Waves and Pattern Dynamics, Springer, pp.3-43, 2018, 978-3-319-78193-8.�10.1007/978-3-319-78193-8_1�. �hal-01703491�

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Angel DuránUniversidad de Valladolid, Spain

Denys DutykhCNRS–LAMA, Université Savoie Mont Blanc, France

Dimitrios MitsotakisVictoria University of Wellington, New Zealand

Peregrine’s system revisited

arXiv.org / hal

Last modified: February 7, 2018

Peregrine’s system revisited

Angel Durán, Denys Dutykh∗, and Dimitrios Mitsotakis

Abstract. In 1967 D. H. Peregrine proposed a Boussinesq-type model for long waves

in shallow waters of varying depth [70]. This prominent paper turned a new leaf in coastal

hydrodynamics along with contributions by F. Serre [72], A. E. Green & P. M. Naghdi

[47] and many others since then. Several modern Boussinesq-type systems stem from

these pioneering works. In the present work we revise the long wave model traditionally

referred to as the Peregrine system. Namely, we propose a modification of the governing

equations which is asymptotically similar to the initial model for weakly nonlinear waves,

while preserving an additional symmetry of the complete water wave problem. This mod-

ification procedure is called the invariantization. We show that the improved system has

well conditioned dispersive terms in the swash zone, hence allowing for efficient and stable

run-up computations.

Key words and phrases: Long dispersive waves; Boussinesq equations; Galilean invari-

ance; wave run-up

MSC: [2010] 76B25 (primary), 76B15, 35Q51, 35C08 (secondary)

PACS: [2010] 47.35.Bb (primary), 47.35.Pq, 47.35.Fg (secondary)

Key words and phrases. Long dispersive waves; Boussinesq equations; Galilean invariance; wave run-

up.∗ Corresponding author.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Mathematical modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Symmetry analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Dimensionless equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Vertical translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Symmetry analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Pressure distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Galilean invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Solitary waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.1 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Numerical discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.1 Finite volume scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2 High order reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.3 Dispersive terms treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.4 Time-stepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6 Landslide generated waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

6.1 Landslide model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

7.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

A. Durán, D. Dutykh & D. Mitsotakis 4 / 42

1. Introduction

Nowadays, Boussinesq-type equations have become the models of choice in the near-shore hydrodynamics. Proposed for the first time in 1871 by J. Boussinesq [15], theseequations have been substantially improved in works by F. Serre (1953) [72], D. H. Pere-

grine (1967) [70], A. E. Green & P. M. Naghdi (1976) [47] and many others∗. Nowadaysit is almost impossible to list all the bibliography on this subject. Since several decennariesresearchers have essentially focused their effort on extending the validity of these modelsfrom shallow waters to intermediate depths [60, 61, 63] under the increasing demand ofthe coastal engineering community. We refer to [20] for a recent reasoned review of thistopic. The derivation of of these equations on flat geometries was reviewed in [55] and thespherical case was covered in [54].

The true success of Boussinesq type equations has to deal with the description ofthe wave breaking phenomenon. Classical Nonlinear Shallow Water Equations (NSWE)predict waves to break too early. Thus, the validity region of NSWE is limited only tothe inner surf zone. The success story of Boussinesq systems begins when they wereshown to model fairly well breaking waves (see [90]). However, the research on robustand efficient numerical methods lags behind the current state of the art in the modeling[8, 11, 41]. Main problems arise from the numerical treatment of the shoreline and thestability of the resulting method. Most of computational algorithms run into numericaltroubles when a sufficiently big amplitude wave reaches the run-up region. These problemsare obviously due to the uncontrolled numerical instabilities coming from the dispersiveterms discretization (see [8]). These difficulties were reported presumably for the first timein P. Madsen et al. (1997) [62] (this emphasis is ours):

“However, to make this technique [slot technique] operational in connectionwith Boussinesq type models a couple of problems call for special attention.[ . . . ] Firstly the Boussinesq terms are switched off at the still water shore-line, where their relative importance is extremely small anyway. Hence inthis region the equations simplify to the nonlinear shallow water equations.”

This extremely pragmatic point of view is still shared nowadays by a number of researchers.However, in our opinion, it is the model which has to decide naturally whether the disper-sion is important or not. Ideally, the treatment of dry areas today should be as simple andnatural as the treatment of shock waves in shock-capturing schemes [81]. In this study wepresent a fully dispersive numerical simulation of a wave run-up on a complex beach wheredispersive terms are present in the entire domain.

The main idea of this study is to revise the original Peregrine system [70]. Someproperties of the complete water wave problem have been lost as a price to pay for themodel simplification. Namely, as for many other models derived by asymptotic methods,we loose the invariance under vertical translations. If no special care is taken, we inevitably

∗The steady version of the celebrated Serre–Green–Naghdi equations can be traced back up to

Lord Rayleigh [59].

Peregrine’s system revisited 5 / 42

h (x)

η (x, t)

H (x, t)

O x

z

Figure 1. Sketch of the fluid domain with a sloping beach.

loose this property, since the asymptotic expansion is performed in a very particular frameof reference (around the mean water level z = 0). However, the full water wave problempossesses this symmetry (cf. [10]).

The model we propose in this study is asymptotically similar to the original systemsince we add only higher order contributions which are formally negligible while greatlyimproving structural properties of the model. Consequently, the linear dispersion relationof the original system is conserved as well. The great improvement consists in dispersiveterms which are better conditioned from the numerical point of view and they fit betterour physical intuition about their relative importance when we approach the shoreline.A similar attempt of improving dispersive terms by adding nonlinear contributions wasalso undertaken recently in [1, 9]. The procedure presented in this study is sometimesreferred to in the literature as the invariantization process. Conservative versions of someNwogu-type systems have been proposed in [7, 44].

The present study is organized as follows. In Section 2 we present some rationale onthe Peregrine system and its invariantization, with particular emphasis on the numericalgeneration of solitary wave solutions of the modified system, which are studied in Section 3.Some elements on the numerical discretization by the finite volume method are given inSection 4. Then, some numerical results are shown in Sections 5 while applications towaves generated due to landslides are presented in 6. Finally, the main conclusions andperspectives of this study are outlined in Section 7.

2. Mathematical modeling

Consider a Cartesian coordinate system in two space dimensions (x, z) to simplifythe notation. The z−axis is taken vertically upwards and the x−axis is horizontal andcoincides traditionally with the still water level. The fluid domain is bounded below by thebottom z = −h (x) and above by the free surface z = η (x, t) . Below we will also need

the total water depth H (x, t)def:= h (x) + η (x, t) . The sketch of the fluid domain is

given in Figure 1. The flow is supposed to be incompressible and the fluid is inviscid. Anadditional simplifying assumption of the flow irrotationality is traditionally made as well.


Remark 1. We would like to underline the fact that in the presence of a free surface thevorticity does not remain zero even if it is so initially. A singularity at the free surface ( e.g.the wave breaking) may lead to vortex sheets creation. However, the water wave theory isnot supposed to hold when a wave breaking event occurs.

Under the previously described physical assumptions, D. H. Peregrine (1967) [70]derived the following system of equations which is valid in the Boussinesq long wave regime:

η t +((h + η) u

)

x= 0 , (2.1)

u t + u u x + g ηx −h

2(h u) xx t +

h2

6u xx t = 0 , (2.2)

where u (x, t) is the depth averaged fluid velocity, g is the gravity acceleration and under-

scripts (u x

def:= ∂u

∂x, η t

def:= ∂η

∂t) denote partial derivatives.

2.1. Symmetry analysis

In this Section we assume the bottom to be flat, i.e. h = d = const > 0 . Other-wise, bathymetry variations will destroy a part of symmetries of the governing equations(2.1), (2.2). The infinitesimal generators of symmetries transformations for the classicalPeregrine system are given here:

X 1 =∂

∂t,

X 2 =∂

∂x,

X 3 = u∂

∂u+ 2 (η + d)

∂

∂η− t

∂

∂t.

It is not difficult to see that the generator X 1 corresponds to time translations:

t = t + ε 1 , x = x , η = η , u = u .

Similarly, the generator X 2 gives translations in space:

t = t , x = x + ε 2 , η = η , u = u .

Finally, the generator X 3 is nothing else but a scaling transformation:

t = e−ε 3 t , x = x , η = e2 ε 3 (d + η) − d , u = eε 3 u .

There are no other symmetry transformations of the classical Peregrine system. If thissystem possessed a Lagrangian structure, we could employ Noether theorem to convertsymmetries to conservation laws [67]. For instance, space translations X 2 correspond to themomentum conservation. The time translations X 1 would yield the energy conservationequation, if we only could apply the Noether theorem. This is one of the reasons why itis widely believed that the classical Peregrine system has no energy functional. However,


using some other complementary methods [13, 22] we were able to compute an additionalconservation law, which can be associated to the energy:

(12u 2 + g (d + η) ln(d + η) − g η − d 2

6u u xx

)

t+

[13u 3 + g u (d + η) ln(d + η) +

d 2

6u x u t −

d 2

6u u t x

]

x= 0 .

The last conservation law can be used, for example, to check the accuracy of numericalschemes over even bottoms for the sake of validation. In some situations additional con-servation laws might be used in theoretical investigations as well.

2.2. Dimensionless equations

Some of our developments below will be more transparent if we work in dimensionlessvariables. The classical long wave scaling is the following:

x ′ def:=

x

ℓ, z ′ def

:=z

h 0, t ′

def:=

g

h 0t , η ′ def

:=η

a, u ′ def

:=u√g h 0

,

where h 0 , a , ℓ are the characteristic water depth, wave amplitude and wave length respec-tively. The following dimensionless numbers are defined from them:

εdef:=

a

h 0

, µ 2 def:=

(h 0

ℓ

) 2

, Sdef:=

ε

µ 2.

Parameters ε and µ 2 measure the wave nonlinearity and dispersion, while the so-calledStokes–Ursell number S measures the relative importance of these effects. In theBoussinesq regime the Stokes–Ursell number is supposed to be of the order of one,i.e. S ∼ 1 . The importance of this parameter is discussed by e.g. F. Ursell (1953)[82]. The Peregrine system (2.1), (2.2) in scaled variables at the order O (ε + µ 2) reads(primes are dropped below for the sake of convenience):

η t +((h + ε η) u

)

x= 0 ,

u t + ε u u x + ηx − µ 2(h

2(h u) xx t −

h 2

6u xx t

)

= O(ε 2 + ε µ 2 + µ 4) ,

where on the right hand side of the last equation we put the order of neglected terms.Since the Stokes–Ursell number S ∼ 1 , we have asymptotic similarity relations in theBoussinesq regime:

ε 2∼ ε µ 2

∼ µ 4 .

2.3. Vertical translations

In this section we examine an important property of the water wave problem — invari-ance under vertical translations (subgroup G 5 in Theorem 4.2, T. Benjamin & P. Olver


(1982) [10]). This transformation is described by the following simple change of variables:

z ← z + d , η ← η − d , h ← h + d , u ← u , (2.3)

where d is some constant. Here again, it is straightforward to check that the mass conser-vation Equation (2.1) remains invariant under transformation (2.3), while Equation (2.2)produces many additional dispersive terms proportional to the constant translation d :


2(h u) xx t +

h 2

6u xx t −

h d

6u xx t −

d 2

3u xx t −

d

2(h u) xx t = 0 .

The reason for this discrepancy is that the coefficient hn (x) (n = 1, 2) in front of thedispersive terms is not invariant under the vertical shift. The right variable to use isthe total water depth H (x, t) = h (x) + η (x, t) which is independent of the chosencoordinate reference frame. Here again, the discrepancy is a result of the asymptoticexpansion around the still water level. Consequently, the derived model is valid only forthis particular choice of the coordinate axis Ox . To make System (2.1), (2.2) frameindependent we shall add higher order nonlinear terms which are asymptotically negligiblebut have important implications in structural properties of the resulting model.

In dimensionless variables the total water depth is expressed as H (x, t) = h (x) +ε η (x, t) . As a corollary, we obtain two asymptotic relations which will be used below:

h = H + O(ε) , hx = H x + O(ε) , H t = O(ε) .

Mathematically it means that the bathymetry function should be completed by an O(ε)term to become invariant under vertical translations. While performing this invariantiza-

tion, we will also recast our model in conservative variables (H, Q) , where Qdef:= H u is

the horizontal momentum. This modification will allow us to employ those numerical meth-ods developed in the literature for the discretization of Nonlinear Shallow Water Equations(NSWE) [27, 35, 40, 91].

The mass conservation Equation (2.1) in the new variables trivially reads:

H t + Qx = 0 , (2.4)

while the momentum conservation Equation (2.2) will require more computations. First ofall, we multiply Equation (2.4) by u , Equation (2.2) by H and add them to have:

(Hu)t +(εH u 2 +

1

2 εH 2)

x− µ 2

(H h

2(h u) xx t −

H h 2

6u xx t

︸︷︷︸

(∗∗)

)

=1

εH hx . (2.5)

In the perspective of writing governing equations in the conservative form, the term (**)has to be transformed using this relation:

u xx ≡(h u

h

)

x x= (h u)

(

2h 2

x

h 3− hxx

h 2

)

− 2hx

h 2(h u) x +

1

h(h u) xx .


Consequently, after simple computations, Equation (2.5) takes the form:

(H u) t +(εH u 2 +

1

2 εH 2)

x− µ 2

(H h

3(h u) xx t +

H

3(h u)(h u) xx x

+H hx

3(h u) x t −

1

3

(H

hh 2

x −1

2H hx x

)(h u) t

)

=1

εH hx .

The last equation is ready for the invariantization process. For illustrative purposes weshow these computations only for the first dispersive term:

µ 2 H h

3(h u) xx t =

µ 2

3H 2 (H u) xx t + O(ε µ 2) =

µ 2

3H 2Qxx t + O(ε µ 2) .

Thus we add again only higher order terms which have no impact onto linear dispersivecharacteristics of the initial system. By proceeding in an analogous manner with all otherdispersive terms and turning back to dimensional variables we obtain the following momen-tum conservation equation:(

1 +1

3H 2

x −1

6HH xx

)

Q t −1

3H 2Qx x t −

1

3HH xQx t

+(Q 2

H+

g

2H 2)

x= g H hx . (2.6)

The system (2.4), (2.6) (that will be called the modified Peregrine system or, in a short-hand notation, the m-Peregrine system) actually has more advantages than being simplyinvariant under two additional transformations. The added value of this invariantizationprocess goes far beyond the initial symmetry consideration. Namely, in this way we extendthe system validity to the run-up process and improve numerical conditioning of dispersiveterms. For the first time Equations (2.4), (2.6) were used and validated for wave run-upproblems in [37].

In natural environments, dispersive effects become gradually less and less importantwhen a wave travels shore-ward to become negligible in the shoreline vicinity. This isthe reason why NSWE can be successfully used to describe to some extent the run-upprocess. This physical observation can be translated into the mathematical language bythe condition that dispersive terms go to zero when the total water depth vanishes. If thiscondition is not fulfilled, numerical instabilities may appear as reported by G. Bellotti

& M. Brocchini (2002) [9]:

“In our attempt to use these equations from intermediate waters up to theshoreline (see Bellotti and Brocchini, 2001) we run into numerical troubleswhen reaching the run-up region, i.e. x > 0 . These problems wereessentially related to numerical instabilities due to the uncontrolled growthof the dispersive contributions (i.e. O(µ 2)-terms).”

The reason for the extended numerical stability of the proposed model is twofold. Firstof all, in the numerical algorithm we have to invert at some point an elliptic operatorwritten over the time derivative in Equation (2.6):

(

1 +1

3H 2

x −1

6HH xx

)

q − 1

3H 2 qx x −

1

3HH x qx = W ,


where W is a known function arising from the advective terms discretization. It turns outthat the resulting linear system is better conditioned if the model is written in terms ofthe total water depth. The second stability advantage comes from the fact that almost alldispersive terms naturally vanish as we approach the shoreline.

Remark 2. It is noted that the same invariantization technique can be used also for thecase of moving bottom bathymetry and for higher dimensions, cf. Section 6.

2.3.1 Symmetry analysis

The symmetries of the m-Peregrine system (over flat bottom) can be computed usingthe standard methods as we did for the classical counterpart in Section 2.1. The dimensionof the symmetry group turns out to be the same as above. The infinitesimal generatorsare given below:

X 1 =∂

∂t,

X 2 =∂

∂x,

X 3 = t∂

∂t+ 2 x

∂

∂x+ 2H

∂

∂H+ 3Q

∂

∂Q.

The generated symmetry transformations are essentially the same. Generator X 1 yieldstime translations:

t = t + ε 1 , x = x , H = H , Q = Q ,

while X 2 gives translations in space:

t = t , x = x + ε 2 , H = H , Q = Q .

Finally, X 3 is a scaling transformation∗:

t = e ε 3 t , x = e 2 ε 3 x , H = e 2 ε 3 H , Q = e 3 ε 3 Q .

2.3.2 Pressure distribution

For some practical applications we need to estimate the pressure field inside the fluid andmore particularly at the bottom. For example, the operational NOAA Tsunami WarningSystem heavily relies on a network of DART buoys detecting tsunami waves by measuringthe pressure at the ocean bottom [12, 79]. In this section we propose a way to reconstructthe pressure field in the whole water column.

In the original work of D. H. Peregrine [70] one can find the following correct asymp-totic expansion for the pressure field:

p = −z + ε η + µ 2(z (h u) x t + 1

2z 2 u x t

)+ O(ε 2 + ε µ 2 + µ 4) . (2.7)

∗Notice, please, that this scaling is different from X 3 given in Section 2.1.


The first two terms on the right hand side correspond to the usual hydrostatic pressurewhile the last two terms are purely non-hydrostatic contributions brought by dispersiveeffects.

However, the original expression (2.7) for the pressure given by the asymptotic expansionmethod has one important drawback. Namely, it satisfies the free surface dynamic bound-ary condition p| z= ε η = 0 only to the leading order. Consequently, the first improvementwe propose is to add some specific higher order terms to recover this property at all orderswe retain in the equation:

p ≈ −z + ε η + µ 2((z − ε η) (h u) xt + 1

2µ 2 (z − ε η) 2 u x t

).

Now we will make a transformation consistent with the modified Peregrine system (2.4),(2.6) which consists in replacing h by its asymptotically equivalent and invariant by verticaltranslations counterpart H in the third term∗ of the last formula:

p ≈ −z + ε η + µ 2((z − ε η) (H u) x t + 1

2µ 2 (z − ε η) 2u x t

).

Finally, if we turn back to the dimensional and conservative variables, the final expressionfor the pressure will take this form:

p

ρ= g (η − z) + (z − η)Qx t +

1

2(z − η) 2

(Q

H

)

x t.

where ρ is the constant fluid density. It is straightforward now to compute the pressurevalue at the bottom by evaluating the last expression at z = −h :

p

ρ

∣∣∣∣z=−h

= g H + H Qx t +1

2H 2(Q

H

)

x t.

The latter can be directly used, for example, to compute synthetic pressure records whichcan be compared with real observations in deep ocean [79].

2.4. Galilean invariance

In the same line of ideas, there is a question of the Galilean invariance of variousBoussinesq-type equations. In this section we check whether the Peregrine system(2.1), (2.2) remains invariant under the Galilean transformation. This issue was alreadyaddressed in the context of some other systems by C. I. Christov (2001) [23].

The procedure is classical. First of all, we assume throughout this section the bottomto be flat h = const. We choose another frame of reference which moves uniformlyrightwards with constant celerity c . Analytically it is expressed by the following change ofvariables:

x ← x − c t , t ← t , η (x, t) ← η (x− c t, t) , u (x, t) ← u (x− c t, t) + c , (2.8)

After some simple computations, one can easily check that the mass conservation Equa-tion (2.1) remains invariant under the Galilean boost (2.8), while Equation (2.2) has an

∗The asymptotic argument holds here since this term is O(µ 2) .


extra term (*):


2(h u) xx t +

h 2

6u xx t +

c h 2

3u xx x

︸︷︷︸

(∗)

= 0 .

Consequently, the Peregrine system in its original form does not possess the very basicGalilean invariance property while the complete water wave problem does (subgroupsG 7, 8 in three dimensions, see Theorem 4.2, T. Benjamin & P. Olver (1982) [10]). Someconsequences of this shortcoming are discussed in Christov (2001) [23].

In order to recover the broken symmetry we propose to modify Equation (2.2) in thefollowing way:


2(h u) xx t +

h 2

6u xx t −

h

3u (h u) xxx = 0 . (2.9)

If we perform the same computations as above, we will see that the modified model (2.1),(2.9) remains invariant under the Galilean boost (2.8). In order to understand betterthis modification, we have to switch to dimensionless variables:

u t + ε u u x + ηx − µ 2(h

2(h u) xx t −

h 2

6u xx t

)

− ε µ 2 h

3u (h u) xxx = 0 .

Now it is clear that we add a higher order O(ε µ 2) nonlinear dispersive term which normallyhas to be omitted according to the philosophy of asymptotic methods. However, we preferto retain it to recover an important physical property of the model — the Galilean

invariance.

Remark 3. Since the term h3u (h u) xx x is a nonlinear dispersive term, it has no effect

onto linear dispersion characteristics of the original model. The same remark applies todevelopments presented below as well.

Consequently, we are able to add a higher order dispersive term to Equation (2.2) whichmakes the system Galilean invariant. The invariantization process in variables (η, u) isstraightforward. However, if we rewrite the modified system in terms of the conservativevariables (H, Q) we loose again the Galilean invariance property. One of the reasons isthat transformation (2.8) is more complex in these variables. For example, the followingchain rules apply:

Q t → Q t − cQx + c (H t − cH x) , Qx → Qx + cH x .

The invariantization of the modified Peregrine system (2.4), (2.6) under the Galilean

symmetry remains an open question. The discussion of the Galilean invariance of a fewother nonlinear dispersive wave systems can be found in [31].

3. Solitary waves

Dispersive wave equations possess an important class of solutions — the Solitary Waves(SW) which result from a balance between nonlinear and dispersive effects [21, 30, 53, 71].


The comprehension of these solutions allows to assess some properties of the dispersivesystem under consideration. We note that analytical SW solutions are not known evenfor the classical Peregrine system [70]. We have not been able to construct closed-form solutions to the m-Peregrine system either. Consequently, we will apply numericalmethods which allow to approximate them accurately [89].

A travelling wave solution has the following form:

H (x, t) ≡ H (X) , Q (x, t) ≡ Q (X) , Xdef:= x − c s t ,

where c s is the wave propagation speed in an inertial frame of reference. After substitutingthis ansatz into the governing Equations (2.4), (2.6), we obtain the following system of twocoupled Ordinary Differential Equations (ODEs):

− c sH′ + Q ′ = 0 , (3.1)

− c s

(

1 +1

3(H ′) 2 − 1

6HH ′′

)

Q ′ +c s

3H 2Q ′′′

+c s

3HH ′ Q ′′ +

(Q 2

H+

g

2H 2) ′

= 0 , (3.2)

where functions H (X) and Q (X) are assumed to be sufficiently smooth, even and decayingto zero along with all their derivatives as |X | → ∞ . Throughout this section we willconsider the wave propagation over a flat bottom, i.e. h ≡ const.

The former Equation (3.1) can be used to eliminate the variable Q (X) from the latterequation. It will be more convenient also to work with the free surface elevation η (X) :

L 0 η = (gh − c 2s) η

′ +c 2s h

2

3η ′′′ +

( c 2s η

2

h + η

) ′

+g

2(η 2) ′ − c 2

s

3(η ′) 3

+c 2s

3(2 h η + η 2) η ′′′ +

c 2s

2(h + η)η ′η ′′ = 0 . (3.3)

Once the free surface elevation η (X) is determined, the velocity can be found from themass conservation (3.1):

u (X) =c s η (X)

h + η (X), Q (X) = c s η (X) . (3.4)

Solitary wave profiles (η (X), u (X)) can be obtained numerically by approximating solu-tions to the differential Equation (3.3) and then using (3.4) to compute the velocity profile.

Several strategies to this end exist in the literature (see [89] and references therein).The one considered here consists of two steps. First, the Newton method is applied to(3.3): from an initial iteration η [0] (X) and if the approximation η [ν] (X) , ν = 0, 1, . . . tothe profile η (X) at the νth iteration is known, then η [ν+1] (X) is obtained by solving theequation

L [ν]∆η [ν] = −L 0 η[ν] , (3.5)

where ∆η [ν] def:= η [ν+1] − η [ν] , L 0 is given by (3.3) and L [ν] is the linearized operator

of Equation (3.3) evaluated at η [ν] (X) .


The second step of our numerical procedure is the discretization of (3.5), which will beinspired by several works of J. Boyd (for more details see [16–19]). For N > 1 and largeL > 0 , the system (3.5) is discretized on the interval

(−L, L

)by the collocation points

x k = −L + (2k + 1) h , h =L

N, k = 0, . . . , N − 1 . (3.6)

For ν = 0, 1, . . . , the approximation η[ν]h to the νth iteration η [ν] is sought in the space

Sh , based on (3.6), of trigonometric interpolation polynomials of the form

Z h (x) =

N − 1∑

j=0

Z j cos( π

2Lj (x + L)

)

.

The discrete version of (3.5) is then as follows. If η [ν] ∈ Sh is known, we search for the

incremental term ∆η[ν]h

def:= η

[ν+1]h − η

[ν]h in Sh , i.e.

∆η [ν] (x) =N − 1∑

j=0

α[ν]j cos

( π

2Lj (x + L)

)

,

and evaluate (3.5) at the collocation points (3.6). This leads to a linear system for the

coefficients α [ν] = (α[ν]0 , . . . , α

[ν]N−1)

⊤ of the form

L[ν]h α [ν] = f [ν] , (3.7)

where the matrix L[ν]h =

(L

[ν]ij

)N − 1

i, j=0and the vector f [ν] = (f

[ν]0 , . . . , f

[ν]N − 1)

⊤ are

computed as:

L[ν]i j = L [ν] cos

( π

2Lj (x + L)

) ∣∣∣x=x i

, f[ν]k = −L 0 η

[ν]∣∣∣X = x k

,

for i, k = 0, . . . , N − 1 . We note that the construction of coefficients f [ν] in (3.7)

requires the computation of derivatives of η[ν]h up to the third order at the points (3.6).

Finally, in order to pass to the next iteration η[ν+1h . Equation (3.7) has to be solved.

The ill-conditioning of the resulting system is treated using the pseudo-inverse techniquecombined with the iterative refinement (see [18, 28, 46, 52] for more details). This methodsolves Equations (3.7) in the least squares sense and the solution has a minimum norm.

The overall iterative process is controlled, in a standard way, by two parameters: (i) amaximum number of iterations and (ii) a tolerance governing the relative error betweentwo consecutive iterations or the residual error:

ε 1 [ν] =‖ η [ν] − η [ν− 1] ‖

‖ η [ν] ‖ , ε 2 [ν] = ‖L 0 η[ν] ‖ , (3.8)

measured in some norm ‖ · ‖ (in the experiments reported below, both the Euclidean

and the maximum norms (l∞) were implemented). Thus, the iteration stops when themaximum number of iterations is attained or when any of the errors (3.8) is below aprescribed tolerance.


3.1. Numerical results

The described above numerical procedure will be tested and used now to compute severaltravelling wave solutions to the m-Peregrine Equations (2.4), (2.6). For the sake ofconvenience, we will solve equations in the dimensionless form which is readily obtained bysetting dimensional constants g = 1 and d = 1 . The tolerance parameter in the controlof the iterations is chosen to be equal to 10−13 . The exact solution to the classical Serre

equations [25, 32, 72] is chosen as the initial approximation at the first iteration.The behaviour of the relative error ε 1 [ν] and absolute error ε 2 [ν] during the iterations

is shown in Figure 2 for two values of the propagation velocity c s = 1.05 and 1.1 .In both cases, the iterations are stopped since the first error drops below the prescribedtolerance. The errors in Figure 2 are measured in the maximum (l∞) norm. The resultsin the Euclidean (l 2) norm are completely similar. We can see that a relatively smallnumber of iterations is needed to achieve the convergence. However, higher values ofthe propagation speed c s lead to higher nonlinearities. Consequently, more iterations areneeded until the convergence is attained. The dependence of the number of iterations onthe speed value c s is illustrated in Figure 3. The metamorphosis of these profiles as wechange gradually the propagation speed c s is shown in Figure 4.

For illustrative purposes we provide several computed amplitudes (free surface elevationand horizontal velocity) of the solitary waves for various values of the propagation speedc s . This speed-amplitude relation is represented graphically in Figure 5. We make alsoa comparison with the 14th order Fenton’s solution for the full water wave problem (formore details see [42, 58]). One can notice a good agreement with the m-Peregrine systemproposed in the previous Section.

4. Numerical discretization

In this section we present briefly the rationale on numerical methods we use to discretizethe system (2.4), (2.6) we derived above: Below we follow the great lines of our previouswork [37].

4.1. Finite volume scheme

We begin our presentation by a discretization of the hyperbolic part of equations (whichare simply the classical nonlinear shallow water equations) and then, in the second time,we discuss the treatment of dispersive terms. The modified Peregrine system (2.4), (2.6)can be formally put under this quasilinear form:

D (v t) + [ f (v) ] x = s (v) , (4.1)


0 5 10 15 20−14

−12

−10

−8

−6

−4

−2

0

N

log

(ε1)

0 5 10 15 20−14

−12

−10

−8

−6

−4

−2

N

log

(ε2)

cs=1.05

cs=1.1

Figure 2. Decimal logarithm of the relative errors defined in (3.8). Relative

difference between two iterations ε 1 [ν] is shown on the left image, while theresidual of the equation is depicted on the right. The convergence is illustratedfor two values of the propagation velocities c s = 1.05 (black solid line) and 1.1

(blue dashed line).

where v , f (v) are the conservative variables and the advective flux function respectively:

v =

(

H

Q

)

, f (v) =

Q

Q 2

H+

g

2H 2

.

The source term s (v) contains the topography effects and D (v t) is the dispersion:

s (v) =

(

0

g H hx

)

, D (v t) =

(

H t(

1 + 13H 2

x − 16HH xx

)

Q t − 13H 2Qxx t − 1

3HH xQx t

)

.

Since the time derivative of the horizontal momentum Q is defined implicitly, we will haveto invert a linear elliptic operator with non-constant coefficients.

The Jacobian of the advective flux f (v) can be easily computed:

A (v) =∂f (v)

∂v=

0 1

g H −(Q

H

) 2 2Q

H

.


1 1.05 1.1 1.15 1.26

8

10

12

14

16

18

20

22

24

26

28

N

cs

Figure 3. Dependence of the number of iterations needed to achieve theconvergence on the solitary wave propagation speed c s .

The Jacobian A (v) has two distinctive eigenvalues:

λ± =Q

H± c s ≡ u ± c s , c s

def:=

√

g H .

The corresponding right and left eigenvectors are provided here:

R =

(

1 1

λ+ λ−

)

, L = R−1 = − 1

2 c s

(

λ− −1−λ+ 1

)

.

Let us fix a partition of R into cells (or finite volumes) C i =[x i− 1

2

, x i+ 1

2

]with cell

centers x i = 12(x i− 1

2

+ x i+ 1

2

) , i ∈ Z . Let ∆x i denotes the length of the cell C i .

Without any loss of generality we assume the partition to be uniform, i.e. ∆x i ≡ ∆x ,∀i ∈ Z . We would like to approximate the solution v (x, t) by discrete values. In orderto do so, we introduce the cell average of v on the cell C i, i.e.

v i (t)def:=

(H i (t), Q i (t)

)=

1

∆x

ˆ

C i

v (x, t) dx .


−15 −10 −5 0 5 10 150

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

x

η

cs=1.2

cs=1.16

cs=1.1

cs=1.06

cs=1.02

(a) η−profile

−15 −10 −5 0 5 10 150

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

x

u

cs=1.2

cs=1.16

cs=1.1

cs=1.06

cs=1.02

(b) u−profile

Figure 4. Solitary wave profiles for various values of the propagation speed c sare superposed on the same image to show the evolution of the shape while

changing this parameter. On the left image we show the free surface profile, whilethe right image depicts the horizontal velocity variable. The lowest curvecorresponds to the smallest values of c s = 1.02 and the highest solution is

obtained for c s = 1.2 .

1 1.05 1.1 1.15 1.20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

η am

p

cs

Modified PeregrineFenton 14th order

(a) Free surface

1 1.05 1.1 1.15 1.20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

ua

mp

cs

Modified Peregrine

(b) Horizontal velocity

Figure 5. Speed-amplitude relation for the m-Peregrine system. On the left

image we show the free surface elevation amplitude and compare it to the 14th

order Fenton’s solution. On the right image we show the horizontal velocityamplitude as a function of the propagation speed c s .


A simple integration of (4.1) over the cell C i leads the following exact relation:

D (v t) i +1

∆x

(

f (v (x i+ 1

2

, t) − f (v (x i− 1

2

, t)))

=1

∆x

ˆ

C i

s (v) d x .

Since the discrete solution is discontinuous at cell interfaces x i+ 1

2

, i ∈ Z , the heart

of the matter in the finite volume method is to replace the flux through cell faces by theso-called numerical flux function:

f (v (x i± 1

2

, t)) ≈ F i± 1

2

(v Li± 1

2

, vRi± 1

2

) ,

where vL,R

i± 1

2

are reconstructions of conservative variables v from left and right sides of

each cell interface. The reconstruction procedure employed in the present study will bedescribed below. Consequently, the semi-discrete scheme takes the form:

D (v t) i +1

∆x

(F i+ 1

2

− F i− 1

2

)= § i , (4.2)

where § i ≈ 1∆x

´

C i

s (v) d x is an approximation of the topographic term on the right-

hand side of (2.6). In the present study we employ the standard hydrostatic reconstruction[2] to obtain a well-balanced scheme. The expression for matrix D will be detailed belowin Section 4.3.

In order to discretize the advective flux f (v) we use the FVCF scheme [45]:

F (v, w) =f(v) + f(w)

2− U (v, w)

f (w) − f (v)

2.

The first part of the numerical flux is centered, the second part is the upwinding introducedthrough the Jacobian sign matrix U (v, w) defined as:

U (v, w) = sign(A (µ)

), sign (A) = R · diag(s+, s−) · L , s± def

:= sign(λ±) .

The average state µ = (µ 1 (v, w), µ 2(v, w) between the left v = (H Li+ 1

2

, uLi+ 1

2

) and the

right w = (H Ri+ 1

2

, uRi+ 1

2

) states∗ is defined as the Roe average:

µ 1 =H L

i+ 1

2

+ H Ri+ 1

2

2, µ 2 =

√

H Li+ 1

2

uLi+ 1

2

+√

H Ri+ 1

2

uRi+ 1

2

√

H Li+ 1

2

+√

H Ri+ 1

2

.

After some simple algebraic computations one can find the following expression for the signmatrix U (v, w) :

U (v, w) =1

2 c

(

s−(µ 2 + c) − s+ (µ 2 − c) s+ − s−

(s+ − s−) (c 2 − µ 22) s+ (µ 2 + c) − s− (µ 2 − c)

)

,

with cdef:=√g µ 1 . We reiterate again that the sign matrix U is evaluated at the average

state µ of left and right values.

∗We do not take here the conservative variables (H, Q) since the reconstruction procedure is more

accurate and robust in physical variables (H, u) .


4.2. High order reconstruction

In order to obtain a higher order scheme in space, we need to replace the piecewiseconstant data by a piecewise polynomial representation. This goal is achieved by variousso-called reconstruction procedures such as MUSCL TVD [56, 83, 84], UNO [51], ENO [50],WENO [88] and many others. In our previous study on Boussinesq-type equations [37],the UNO2 scheme showed a good performance with low dissipation in realistic propagationand run-up simulations.

Remark 4. In TVD schemes the numerical operator is required (by definition) not toincrease the total variation of the numerical solution at each time-step. It follows that thevalue of an isolated maximum may only decrease in time which is not a good property forthe simulation of coherent structures such as solitary waves. The non-oscillatory UNO2scheme, employed in our study, is only required to diminish the number of local extrema inthe numerical solution. Unlike TVD schemes, UNO schemes are not constrained to dampthe values of each local extremum at every time-step.

The main idea of the UNO2 scheme is to construct a non-oscillatory piecewise-parabolicinterpolant Q (x) to a piecewise smooth function v (x) (see [51] for more details). On eachsegment containing the face x i+ 1

2

∈ [x i, x i+1 ] , the function Q (x) = q i+ 1

2

(x) is locally

a quadratic polynomial and wherever v (x) is smooth we have:

Q (x) − v (x) = O(∆x 3),dQ

dx(x ± 0) − dv

dx= O(∆x 2) .

Also Q (x) should be non-oscillatory in the sense that the number of its local extrema doesnot exceed that of v (x) . Since q i+ 1

2

(x i) = v i and q i+ 1

2

(x i+1) = v i+1 , it can be

written in the form:

q i+ 1

2

(x) = v i + d i+ 1

2

v · x − x i

∆x+ 1

2D i+ 1

2

v · (x − x i)(x − x i+1)

∆x 2,

where d i+ 1

2

vdef:= v i+1 − v i and D i+ 1

2

v is closely related to the second derivative of the

interpolant since D i+ 1

2

v = ∆x 2 q ′′

i+ 1

2

(x) . The polynomial q i+ 1

2

(x) is chosen to be one

the least oscillatory between two candidates interpolating v (x) at (x i− 1, x i, x i+1) and(x i, x i+1, x i+2) . This requirement leads to the following choice of D i+ 1

2

v :

D i+ 1

2

vdef:= minmod

(D i v, D i+1 v

),

with

D i v = v i+1 − 2 v i + v i− 1 , D i+1 v = v i+2 − 2 v i+1 + v i ,

and minmod (x, y) is the usual min mod function defined as:

minmod (x, y) =1

2(sign(x) + sign(y)) ·min(|x | , | y |) .


To achieve the second order O(∆x 2) accuracy it is sufficient to consider piecewise linearreconstructions in each cell. Let L (x) denote this approximately reconstructed functionwhich can be written in this form:

L (x) = v i + s i ·x − x i

∆x, x ∈

[x i− 1

2

, x i+ 1

2

].

To make L (x) a non-oscillatory approximation we use the parabolic interpolation Q (x)constructed below to estimate the slopes s i within each cell:

s i = ∆x ·minmod(dQ

dx(x i − 0),

dQ

dx(x i + 0)

)

.

In other words, the solution is reconstructed on the cells while the solution gradient isestimated on the dual mesh as it is often performed in more modern schemes [3, 4]. A briefsummary of the UNO2 reconstruction can be also found in [37].

4.3. Dispersive terms treatment

In this section we explain how we treat the dispersive terms of the m-Peregrine system(2.4), (2.6). Here again, we follow in great lines our previous study [37]. The followingsecond order O(∆x 2) approximations are used to discretize the dispersive terms arising inmatrix D (v t) :

1

∆x

ˆ

C i

[

1 +1

3H 2

x −1

6H H xx

]

Q t d x ≈

(

1 +1

3

(H i+1 − H i− 1

2∆x

) 2 − 1

6H i

H i+1 − 2H i + H i− 1

∆x 2

)

(Q t) i ,

1

∆x

ˆ

C i

1

3HH x Qx t d x ≈

1

3H i

H i+1 − H i− 1

2∆x

(Q t) i+1 − (Q t) i− 1

2∆x,

1

∆x

ˆ

C i

1

3H 2 Qxx t d x ≈

1

3H 2

i

(Q t) i+1 − 2 (Q t) i + (Q t) i− 1

∆x 2.

Given the previous discretizations we obtain the following semi-discrete scheme:

dH i

dt+

1

∆x

(F

(1)

i+ 1

2

− F(1)

i− 1

2

)= 0 , (4.3)

LdQ i

dt+

1

∆x

(F

(2)

i+ 1

2

− F(2)

i− 1

2

)= S (v) . (4.4)

The matrix D defined above in Equation (4.2) can be expressed in terms of the matrix L :

Ddef:=

(

I 0

0 L

)

,

where I is the identity matrix.


Consequently, in order to obtain the fully discrete scheme from Equations (4.3), (4.4)we have to invert a system of linear equations with the tridiagonal matrix L . It can bedone efficiently with linear complexity. We note that on dry cells the matrix L becomessimply the identity matrix since H i ≡ 0 in that regions. We reiterate again that wedo not switch off the dispersive terms at some empirically chosen depth. It is the wavepropagation physics which governs the magnitude of dispersive terms and thus, will decidewhether they are important or not.

4.4. Time-stepping

We assume that the linear system of equations is already inverted leading to a systemof ODEs of the form:

v t = N (v, t) , v (0) = v 0 .

In order to solve numerically the last system of equations, we apply the Bogacki–Shampine

method proposed in [14]. It is a Runge–Kutta scheme of the third order with four stages.It has an embedded second order method which is used to estimate the local error and thus,to adapt the time-step size. Moreover, the Bogacki–Shampine method enjoys the FirstSame As Last (FSAL) property so that it needs approximately three function evaluationsper step. This method is also implemented in the ode23 function in Matlab [73]. The onestep of the Bogacki–Shampine method is given by:

k 1 = N (v (n), tn) ,

k 2 = N (v (n) + 12∆tn k 1, tn + 1

2∆t) ,

k 3 = N (v (n)) + 34∆tn k 2, tn + 3

4∆t) ,

v (n+1) = v (n) + ∆tn(29k 1 + 1

3k 2 + 4

9k 3

),

k 4 = N (v (n+1), tn +∆tn) ,

v(n+1)2 = v (n) + ∆tn

(424

k 1 + 14k 2 + 1

3k 3 +

18k 4

).

Here v (n) ≈ v (tn) , ∆t is the time-step and v(n+1)2 is a second order approximation to

the solution v (tn+1) , so the difference between v (n+1) and v(n+1)2 gives an estimation of

the local error. The FSAL property consists in the fact that k 4 is equal to k 1 in the nexttime-step, thus saving one function evaluation.

If the new time-step ∆tn+1 is given by ∆tn+1 = ρn ∆tn , then according to H211b

digital filter approach [74, 75], the proportionality factor ρn is given by:

ρn =( δ

εn

)β 1( δ

εn− 1

)β 2

ρ−αn− 1 , (4.5)

where εn is a local error estimation at time-step tn and constants β 1 , β 2 and α are definedas:

α =1

4, β 1 =

1

4 p, β 2 =

1

4 p.

The parameter p is the order of the scheme and p = 3 in our case.


Remark 5. The adaptive strategy (4.5) can be further improved if we regularize the factorρn before computing the next time-step ∆tn+1 :

∆tn+1 = ρn ∆tn , ρn = ω (ρn) .

The function ω (ρ) is called the time-step limiter and should be smooth, monotonicallyincreasing and should satisfy the following conditions:

ω (0) < 1 , ω (+∞) > 1 , ω (1) = 1 , ω ′ (1) = 1 .

One possible choice was suggested in [75]:

ω (ρ) = 1 + κ arctan(ρ − 1

κ

)

.

In our computations the parameter κ is set to 1 .

Several validations of the above presented numerical scheme, including the convergencetests, run-up simulations as well as the comparison with experimental data [76, 90] can befound in our previous numerical study [37]. Here we make a step forward in the applicationof the proposed numerical model to practical coastal engineering problems.

5. Numerical results

Using the numerical method described in the preceding section, we can perform somesimulations of the wave run-up onto a plane beach. Consider a setup schematically depictedin Figure 1. The bathymetry defined on a segment

[a, c

]is composed of two regions:

constant depth region z = −d 0 , for x ∈[a, b

]and the constant slope region z =

−d 0 + x tan(δ) , x ∈[b, c

]. We will solve numerically a Boundary Value Problem (BVP).

Namely, on the right end (x = c) we impose the wall boundary condition u|x= c = 0 ,while on the left boundary (x = a) we are given by the incident wave height. In thepresent study we will consider the run-up of a monochromatic periodic wave entering fromthe left side (see Figure 1):

H 0 (t) = d 0 + A sin(ω t) .

The computational domain is discretized into N = 500 equal control volumes. The time-step value is automatically chosen by the time-stepping algorithm. The values of variousphysical parameters are given in Table 1.

Remark 6. The rigorous imposing of an incident wave boundary condition in the contextof various dispersive wave equations is essentially an open question. However, for the m-Peregrine system under consideration, we found an operational solution based on thehyperbolic part of these equations. The general method is described in [68]. The numericalflux through the first left face x = a is found by considering incoming characteristics andis given by this formula:

F (x = a, t) =

(

H 0 (t) u 0

H 0 (t) u20 + g

2H 2

0 (t)

)

, u 0def:= u 1 +

(1 − H 1

H 0

)√

g H 1 ,


Undisturbed water depth, d0 1

Gravity acceleration, g 1

Incident wave amplitude, A 0.3

Incident wave frequency, ω 0.8

Final simulation time, T 29.0

Left boundary coordinate, a -8

Transition coordinate between regions, b 0

Right boundary coordinate, c 16

Beach slope, tan(δ) 0.14

Table 1. Values of various parameters used in convergence tests.

−5 0 5 10 15

−1

−0.5

0

0.5

1

x

η(x,t

)

Free surface elevation at t = 15.00

mPeregrine

NSWE

Bottom shape

Figure 6. Free surface snapshot at t = 15 . The blue solid line corresponds tothe m-Peregrine system, the black dashed line refers to NSWE and the reddot-dashed line shows the bottom.

where (H 1, u 1) are the reconstructed physical variables on the left face from the fluid do-main. Our numerical tests presented below demonstrate the robustness and efficiency ofthis approach.

The afore-described situation is simulated with the modified Peregrine system (2.4),(2.6), but also with classical nonlinear shallow water equations (NSWE) [38, 40, 91]. Thecomparative results of this simulation are presented in Figures 6 – 9. We underline thatno friction terms are considered in this study. The numerical results we present are basedonly on mathematical models described above.

During the initial stages, which are not shown in figures for the sake of manuscriptcompactness, we see the periodic wave entering into the computational domain. The non-dispersive solution is much steeper and first shock waves start to form. Then, the wavecontinues its propagation towards the shore. During the propagation and run-up processes,


−5 0 5 10 15

−1

−0.5

0

0.5

1

x

η(x,t

)


mPeregrine

NSWE

Bottom shape

Figure 7. Free surface snapshot at t = 18 . The blue solid line corresponds to

the m-Peregrine system, the black dashed line refers to NSWE and the reddot-dashed line shows the bottom.

−5 0 5 10 15

−1

−0.5

0

0.5

1

x

η(x,t

)


mPeregrine

NSWE

Bottom shape

Figure 8. Free surface snapshot at t = 20 . The blue solid line corresponds tothe m-Peregrine system, the black dashed line refers to NSWE and the red

dot-dashed line shows the bottom.

the solution to the m-Peregrine system is always behind the hyperbolic wave and this isdue to dispersive effects which make the wave propagation speed closer to its physical value.The run-up process starts about t = 15 and it can be seen in Figure 6. The developmentof this process is shown in Figures 7 – 9. Both waves about their maximum run-up heightare depicted in Figure 9. It is interesting to observe a shock-like wave formed by the m-Peregrine system near the shore in Figure 9. It shows that in the shallowest regions thewave dynamics is governed essentially by nonlinear effects. This transition is naturally andautomatically captured by our system without adding any ad-hoc parameters.


−5 0 5 10 15

−1

−0.5

0

0.5

1

x

η(x,t

)


mPeregrine

NSWE

Bottom shape

Figure 9. Free surface snapshot at t = 25 . The blue solid line corresponds tothe m-Peregrine system, the black dashed line refers to NSWE and the red

dot-dashed line shows the bottom.

6. Landslide generated waves

Extreme water waves can become an important hazard in coastal areas. Main geophysicalmechanisms include underwater earthquakes and landslides. The former genesis mechanismhas been intensively investigated since the Tsunami Boxing Day [6, 33, 36, 39, 64–66, 77].The list of references is far from being exhaustive. In this section we focus on the lattermechanism – the underwater landslides which can cause some considerable damage in thegenesis region. In general, the wavelength of landslide generated waves is much smallerthan the length of transoceanic tsunamis. Consequently, the dispersive effects might beimportant. This consideration explains why we opt for a dispersive m-Peregrine modelwhich is able to simulate the propagation and run-up of weakly nonlinear weakly dispersivewater waves on nonuniform beaches.

Most of the landslide models which are currently used in the literature can be convention-ally divided into three big categories. The first category contains the simplest models wherethe landslide shape and its trajectory are known a priori [57, 78, 80]. Another approachconsists in assuming that the landslide motion is translational and the sliding mass followsthe trajectory of its barycenter. The governing equation of the center of mass is obtainedby projecting all the forces, acting on the slide, onto the horizontal direction of motion[29, 48, 85]. Finally, the third category of models describes the slide-water evolution as atwo-layer system, the sliding mass being generally formulated by a Savage–Hutter typemodel [43]. Taking into account all the uncertainties which exist in the modeling of thereal-world events, we choose in this chapter to study the intermediate level (i.e. the sec-ond category) which corresponds better to the precision of the available data in real-worldsituations. The chosen landslide model will be detailed below in Section 6.1.


The original derivation of the Peregrine system [70] assumes that the bottom is sta-tionary in time, i.e. z = −h (x) . However, in order to simulate the wave generationprocess by bottom motion we need to include the time dependence into the bathymetrydefinition [33, 34]. The bottom dynamics has been included into the Peregrine systemderivation by T. Wu [86, 87]:

η t +((h + η) u

)

x= − h t ,


2(h u) xx t +

h 2

6u xx t =

1

2h hx t t

︸︷︷︸

(∗)

,

where the new term due to the bottom motion is marked with sign (*). By repeating thesame invariantization process as above, the system written in conservative variables andwith moving bottom can be straightforwardly derived:

H t + Qx = 0 , (6.1)

(1 +

1

3H 2

x −1

6HH xx

)Q t −

1

3H 2Qx x t −

1

3HH xQx t +

(Q 2

H+

g

2H 2)

x

= g H hx +1

2H2 hx t t . (6.2)

The bottom motion enters into the momentum balance Equation (6.2) through the sourceterm 1

2H d x t t . The mass conservation Equation (6.1) keeps naturally its initial form. We

underline that the linear dispersion relation of the m-Peregrine system (6.1), (6.2) isidentical with that the original Peregrine model [70] since these models differ only innonlinear terms and the source terms do not enter into the dispersion relation analysis. Thenumerical scheme described in Section 4 is applied to the moving bottom m-Peregrine

system (6.1), (6.2) without any modification. The new source term is just projected ontocell centers since the function h (x, t) is prescribed by the bathymetry, the landslide shapeand trajectory.

Remark 7. Following the same invariantization one can derive the two-dimensional mod-ified Peregrine system including moving bottom topography:

H t +∇ ·Q = 0 (6.3)

Q t +∇ ·(

Q ⊗ Q

H+

g

2H 2 I

)

− P (H, Q ) = g H ∇ h +H2

2∇h tt , (6.4)

where

P (H, Q ) =H 2

2∇(∇·Q t) −

H2

6∆Q t −

( |∇H|23− H ∆H

6

)

Q t +1

3H ∇H ·∇Q t .

It is noted that in this case H depends on ( x, y, t ) and Q = H × (u, v)T with u ( x, y, t )and v ( x, y, t ) being the depth-averaged velocity horizontal components of the fluids ve-locity in the directions x and y respectively. This system again contains some high-order


correction terms in the source terms that can be simplified without affecting the invarianceof vertical translations.

6.1. Landslide model

In this section we briefly present a model of an underwater landslide motion. This processhas to be addressed carefully since it determines the subsequent formation of water waves.In this study we will assume the moving mass to be a solid quasi-deformable body with aprescribed shape and known physical properties that preserves its mass and volume. Underthese assumptions it is sufficient to compute the trajectory of the barycenter x = x c (t) todetermine the motion of the whole body. In general, only uniform slopes are considered inthe literature in conjunction with this type of landslide models [24, 29, 48, 69, 85]. However,a novel model, taking into account the bottom geometry and curvature effects, has beenrecently proposed [6]. Hereafter we will follow in great lines this study.

The static bathymetry is prescribed by a sufficiently smooth (at least of the class C 2)and single-valued function z = −h 0 (x) . The landslide shape is initially prescribed by alocalized in space function z = ζ 0 (x) . For example, in this study we choose the followingshape function:

ζ 0 (x) = A sech(k (x − x 0)

), (6.5)

where the parameter A is the maximum slide height, k is inversely proportional to the slidelength and x 0 is the initial position of its barycenter. Obviously, the model descriptiongiven below is valid for any other reasonable shape.

Since the landslide motion is translational, its shape at time t is given by the functionz = ζ (x, t) = ζ 0 (x − x c (t)) . Recall that the landslide center is located at the point withabscissa x = x c (t) . Then, the impermeable bottom for the water wave problem can beeasily determined at any time by simply superposing the static and dynamic components:

z = −h (x, t) = −h 0 (x) + ζ (x, t) .

To simplify the subsequent presentation, we introduce the classical arc-length parametriza-tion, where the parameter s = s (x) is given by the following formula:

s = L (x) =

ˆ x

x 0

√

1 + (h ′

0(ξ))2 d ξ . (6.6)

The function L (x) is monotonic and can be efficiently inverted to turn back to the originalCartesian abscissa x = L−1 (s) . Within this parametrization, the landslide is initiallylocated at point with the curvilinear coordinate s = 0 . The local tangential direction isdenoted by τ and the normal by n .

The landslide motion is governed by the following differential equation obtained by astraightforward application of Newton’s second law:

md2 s

dt 2= F τ (t) ,


where m is the mass and F τ (t) is the tangential component of the forces acting on themoving submerged body. In order to project the forces onto the axes of local coordinatesystem, the angle θ (x) between τ and Ox can be easily determined:

θ (x) = arctan(h ′

0 (x)).

Let us denote by ρw and ρ ℓ the densities of the water and sliding material correspondingly.If V is the volume of the slide, then the total mass m is given by

mdef:= (ρ ℓ + cw ρw) V ,

where cw is the added mass coefficient [5]. A portion of the water mass has to be addedsince it is entrained by the underwater body motion. The volume V can be computed as

V = W · S = W

ˆ

R

ζ 0 (x) d x ,

where W is the landslide width in the transverse direction. The last integral can becomputed exactly for the particular choice (6.5) of the landslide shape to give

V =1

2ℓ AW .

The total projected force F τ acting on the landslide can be conventionally representedas a sum of two different kind of forces denoted by F g and F d :

F τ = F g + F d ,

where F g is the joint action of the gravity and buoyancy, while F d is the total contributionof various dissipative forces (to be specified below). The gravity and buoyancy forces actin opposite directions and their horizontal projection F g can be easily computed:

F g (t) = (ρ ℓ − ρw)W g

ˆ

R

ζ (x, t) sin(θ (x)

)d x .

Now, let us specify the dissipative forces. The water resistance to the motion force F r isproportional to the maximal transversal section of the moving body and to the square ofits velocity:

F r = −12c d ρw AW σ (t)

(ds

dt

) 2

,

here c d is the resistance coefficient of the water and σ (t)def:= sign

(dsdt

)

. The coefficient

σ (t) is needed to dissipate the landslide kinetic energy independently of its direction ofmotion. The friction force F f is proportional to the normal force exerted on the body dueto the weight:

F f = −c f σ (t)N (x, t) .

The normal force N (x, t) is composed of the normal components of gravity and buoyancyforces but also of the centrifugal force due to the variation of the bottom slope:

N (x, t) = (ρ ℓ − ρw) gW

ˆ

R

ζ (x, t) cos(θ (x)

)d x + ρ ℓW

ˆ

R

ζ (x, t) κ (x)(ds

dt

) 2

d x ,


where κ (x) is the signed curvature of the bottom which can be computed by the followingformula:

κ (x) =h ′′

0 (x)(1 + (h ′

0 (x))2) 3

2

.

We note that the last term vanishes for a plane bottom since κ (x) ≡ 0 in this particularcase.

In order to dissipate more energy along the landslide trajectory if it is needed, we com-plete our model by two supplementary viscous terms:

F d = −c v

ds

dt− c b

ds

dt

∣∣∣∣

ds

dt

∣∣∣∣,

where c v and c b are some prescribed constants. The first term c v represents the internalenergy loss inside the sliding material. The second term c b accounts for the dissipation inthe boundary layer between the landslide and the solid bottom.

Finally, if we sum up all the contributions of described above forces, we obtain thefollowing second order differential equation:

(γ + cw)Sd2 s

dt 2= (γ − 1) g

(

I 1 (t) − c f σ (t) I 2 (t))

− σ (t)(

c f γ I 3 (t) +1

2c dA

)(ds

dt

) 2

− c v

ds

dt− c b

ds

dt

∣∣∣∣

ds

dt

∣∣∣∣, (6.7)

where γdef:= ρ ℓ

ρw> 1 is the ratio of densities and integrals I 1, 2, 3 (t) are defined as:

I 1 (t) =

ˆ

R

ζ (x, t) sin(θ (x)

)d x ,

I 2 (t) =

ˆ

R

ζ (x, t) cos(θ (x)

)d x ,

I 3 (t) =

ˆ

R

ζ (x, t) κ (x) d x .

Note also that Equation (6.7) was simplified by dividing both sides by the width value W .In order to obtain a well-posed initial value problem, Equation (6.7) has to be completedby two initial conditions:

s (0) = 0 , s ′ (0) = 0 .

From Equation (6.7) it follows that the motion can start only if this condition is fulfilled[6]:

I 1 (0) − c f I 2 (0) =

ˆ

R

ζ 0 (x)[

sin(θ (x)

)− c f cos

(θ (x)

)]

d x > 0 .

In order to solve numerically Equation (6.7) we employ the same Bogacki–Shampine

3rd order Runge–Kutta scheme that we used to approximate the Boussinesq Equations(6.1), (6.2). The integrals I 1, 2, 3 (t) are computed with the trapezoidal rule. Once thelandslide trajectory s = s (t) is found, Equation (6.6) is used to find its motion x = x (t)in the initial Cartesian coordinate system.


6.2. Numerical results

Consider a one-dimensional physical domain I =[a, b

]=

[−120, 120

]which is

divided into N equal control volumes. This domain is composed of three regions: the leftand right curvilinear sloping beaches which surround a generation region of a deformedparabolic shape. Specifically, the static bathymetry function d 0 (x) is given by the followingexpression:

d 0 (x) = −κ(x 2 − c 2

)+ A 1 e

−k 1 (x − x 1) 2 + A 2 e−k 2 (x − x 2) 2 .

Basically, this function represents a parabolic bottom profile deformed by two underwaterbumps. We made this nontrivial choice in order to illustrate better the advantages of ourlandslide model, which was designed to handle general non-flat bathymetries. The valuesof all physical and numerical parameters are given in Table 2. The bottom profile alongwith landslide trajectory for these parameters are depicted in Figure 10. The landslidemotion starts from the rest position under the action of the gravity force. We simulate itsmotion along with the free surface waves up to time T = 150.0 s. As it is expected, thelandslide remains trapped between two underwater bumps in its final equilibrium position.The speed and acceleration of the slide barycenter during the simulation are represented inFigure 11. We note the discontinuities in the acceleration record which correspond to thetime moments when the velocity changes its sign. We insist that this behaviour is intrinsicto the landslide model in use where the dissipative terms show the discontinuous behaviourat turning points.

One of the important parameters in shallow water flows is the Froude number, de-fined as the ratio between the characteristic fluid velocity to the gravity wave speed. Wecomputed also this parameter along the landslide trajectory:

Fr (t)def:=

|x ′

c (t) |√

g d(x c (t), t)

) .

The result is presented in Figure 12. We can see that in our case the slide motion remainssub-critical as it is the case in most real world situations [49].

In order to measure the free surface elevations due to the underwater landslide, weinstalled four numerical wave gauges located at x = x 0 , −50.0 , 0.0 and 50.0 . Thesynthetic wave records are presented in Figure 13. One can see that the biggest quantityof primary interest is the wave run-up onto left and right beaches surrounding the fluiddomain. This quantity is estimated numerically using the previously described algorithm.The shoreline motion is represented in Figure 14. One can see that the landslide scenariounder consideration produces much higher run-up values on the beach opposite to the slopewhere the sliding process takes place. Finally, in order to illustrate the energy transferprocess from the landslide motion to the fluid layer, we show the evolution of both energiesduring the generation process in Figure 15. We recall that the fluid potential, kinetic and


Parameter Value

Gravity acceleration, g 1.0

Parabolic bottom flatness coefficient, κ 1.5× 10−3

Initial shoreline position, c 100.0

Underwater bump amplitude, A 1 2.8

Underwater bump amplitude, A 2 −4.8Bump characteristic steepness, k 1 0.008

Bump characteristic steepness, k 2 0.003

Bump center position, x 1 −60.0Bump center position, x 2 0.0

Number of control volumes, N 2500

Slide amplitude, A 0.5

Characteristic slide inverse length, k 0 0.16

Initial slide position, x 0 −85.0Added mass coefficient, cw 1.0

Water drag coefficient, c d 1.0

Friction coefficient, c f tan 2◦

Ratio between water and slide densities, γ 2.0

Boundary layer dissipation coefficient, c b 0.0035

Internal friction coefficient, c v 0.0045

Final simulation time, T 150.0

Table 2. Values of various parameters used in the numerical computations.

total energies are defined correspondingly as

Π (t)def:=

1

2

ˆ

R

g η 2 d x , K (t)def:=

1

2

ˆ

R

(d + η) u 2 d x , E (t)def:= Π (t) + K (t) .

The landslide kinetic energy is readily obtained from the differential Equation (6.7):

K ℓ (t)def:=

1

2(γ + cw)S

(ds

dt

) 2

.

Our computation shows that only about 10% of the landslide energy is transmitted to thewave. This estimation is in complete accordance with values reported by Harbitz et al.[49].

7. Discussion

Below we outline the main conclusions and perspectives of our study.


−100 −50 0 50 1000

50

100

150

xc(t)

t

Barycenter trajectory

−100 −50 0 50 100−15

−10

−5

0

5

x

-d(x)

Bathymetry and landslide position

Landslide position

Static bottom profile

Figure 10. Bathymetry profile and the landslide trajectory for the parameters

given in Table 2. The initial landslide position is shown on the lower image withthe red dashed line (- - -).

7.1. Conclusions

In the present study we revisited the celebrated Peregrine system for long waves prop-agation. Namely, our primary goal was to undertake a series of equivalent transformationswhich do not modify lower order dispersive terms O(µ 2) , while extending the model sta-bility and validity up to the shoreline. Moreover, the resulting governing equations possessan additional symmetry of the complete water wave problem which were broken as a re-sult of the asymptotic expansion. Hence, our model remains invariant under the verticaltranslation (subgroup G 5 in Theorem 4.2, T. Benjamin & P. Olver (1982) [10]). Theapplication of the invariantization process presented in this study can be extended to anyother system of Boussinesq type. It can be viewed as a post treatment procedure to beapplied after the derivation of a particular model. The Peregrine system was chosen forillustrative purposes due to its importance and popularity in the water wave community.


0 50 100 150−1

0

1

2

t

v c(t)

Landslide speed

0 50 100 150−0.1

−0.05

0

0.05

0.1

0.15

t

ac(t)

Landslide acceleration

Figure 11. Landslide speed and acceleration along its trajectory.

0 50 100 1500

0.1

0.2

0.3

0.4

0.5

0.6

Local Froude number

t

Fr(t)

Figure 12. Local Froude number computed along the slide motion.


0 50 100 150−0.4

−0.2

0

0.2

t

η(t)/A

x = −85.0

0 50 100 150

−1

0

1

t

η(t)/A

x = −50.0

0 50 100 150−1

0

1

t

η(t)/A

x = 0.0

0 50 100 150−1

0

1

t

η(t)/A

x = 50.0

Figure 13. Synthetic wave gauge records at four different locations. Note the

different vertical scales on various images. Wave gauges are located atx = x 0 = −85.0 , −50.0 , 0.0 , 50.0 from the top correspondingly. The waveamplitude is relative to the landslide amplitude.

Of course, this system possesses also several nice properties which explain its wide usagein applications.

The developments made in this study are illustrated with several examples. First of all,we proposed an efficient numerical method to construct travelling wave solutions. Somecomparisons with the classical Nonlinear Shallow Water equations (NSWE) were presentedfor the wave run-up problem onto a plane beach. The effect of dispersive terms is exempli-fied. In this study we also presented a model of a landslide motion over general curvilinearbottoms. This model takes into account the effects of bottom curvature, generally ne-glected in the literature [29, 48, 69, 85]. Despite the inclusion of some new physical effects,the considered slide model is computationally inexpensive and can be potentially used inmore operational context. We tested the m-Peregrine model on this more realistic caseof the wave generation by an underwater landslide. The coupling with the m-Peregrine

equations was done through the time-dependent bathymetry. Wave run-up records on non-flat beaches were computed. The proposed technique can be directly applied to perform alandslide hazard effects in real-world situations.


0 50 100 150−0.5

0

0.5

t

Ra(t)/A

Run−up on the left beach

0 50 100 150

−2

−1

0

1

2

3

t

Rb(t)/A

Run−up on the right beach

Figure 14. Wave run-up heights onto left and right non-flat beaches during thesimulation.

7.2. Perspectives

In the present manuscript we focused on the two-dimensional (2D) physical problem,which became a one-dimensional (1D) mathematical problem thanks to the eliminationof explicit dependence on the vertical coordinate (1DH). In future works we are going tofocus on the generalization of the m-Peregrine to the 2DH situation with two horizontaldirections. There is another question which can be asked even in the 1D case — the energyconservation issue. So far, a successful response to this question has been brought in thevariational framework [26].


0 10 20 30 40 50 60 700

1

2

3

4

5

6

t

E(t)

Water wave energy

Kinetic energy

Potential energy

Total energy

0 10 20 30 40 50 60 700

10

20

30

40

50

60

t

Kl(t)

Kinetic energy of the landslide

Slide energy

Figure 15. Fluid and landslide energies evolution during the wave generation process.

Acknowledgments

D. Dutykh & A. Durán acknowledge the support from project MTM2014-54710-P enti-tled “Numerical Analysis of Nonlinear Nonlocal Evolution Problems” (NANNEP). D. Mit-

sotakis was supported by the Marsden Fund administered by the Royal Society ofNew Zealand.

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A. Durán: Departamento de Matemática Aplicada, E.T.S.I. Telecomunicación, Campus

Miguel Delibes, Universidad de Valladolid, Paseo de Belen 15, 47011 Valladolid, Spain

E-mail address : [email protected]

URL: https://www.researchgate.net/profile/Angel_Duran3/

D. Dutykh: Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, 73000

Chambéry, France and LAMA, UMR 5127 CNRS, Université Savoie Mont Blanc, Campus

Scientifique, F-73376 Le Bourget-du-Lac Cedex, France


URL: http://www.denys-dutykh.com/

D. Mitsotakis: Victoria University of Wellington, School of Mathematics, Statistics

and Operations Research, PO Box 600, Wellington 6140, New Zealand


URL: http://dmitsot.googlepages.com/

angel durán, denys dutykh, dimitrios mitsotakis

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